Is it possible to host a Babel language service outside of Visual Studio so that it can be packaged into an end user tool, rather than just a developer helper?
Certainly. Are you the babel language developer? Or are you just an user who is wanting to leverage it in an different tool? If you are not the developer I'm guessing you will have some real trouble doing it, licensing and otherwise. A visual studio language service is usually pretty integrated with Visual Studio so I'm guessing you won't be able to re-use much of that but for compiling languages it usually boils down to an MSBuild task, which you can definitely leverage from other tools easily. Look in the babel project file and find out where it is importing its tasks from, then find that task and which assemblies it uses for those tasks. You could easily add a reference to that assembly or leverage MSBuild to execute that compile task.
